by tension upon em, while the weight at c tends to throw 2w" upon it by thrust, as seen above. The result is, that em sustains 1w" by tension, and cm, 1w" by thrust, while co sustains the whole weight at c, in addition to 1w" from e.

Again, a weight (w) at f, tends to throw 2" upon fl, to act by tension; but as fl is already occupied by 4w" acting by thrust, ƒ is obliged to depend entirely upon fj; at the same time turning back 2", and reducing the weight previously on If, by that amount, or, to 2w".

In like manner, a weight applied at g, finds gk sustaining 6w" by thrust, whereby it is prevented from sending 1w" (due from it at a), to the left, through tension of gk. Hence, the whole weight at gis sustained by jg, and the weight acting on kg is reduced to 5w".

XLV. Thus we see that each diagonal (except oc and fj, excluded by hypothesis), is liable to compression from weights at certain points, and tension from weights at other points; and, it is manifest that the greater stress of either kind, on each diagonal, is when all the weights are on the truss, which tend to produce upon it one kind of stress, and none of those which tend to produce the opposite stress.

Hence, if we place the numbers 1, 2, 3, &c., over the diagram, as in case of Fig. 12. [XXXIX], it is clear that only alternate weights act upon the same system of diagonals; that only weights under the odd numbers. 1, 3 and 5 act upon diagonals meeting the lower chord at points under those numbers; and so of the weights under the even numbers 2, 4 and 6. We therefore form a second series offigures under the first, by placing under each odd number, the sum of that number and

all the preceding odd numbers, and under each even number, the sum of that and all preceding even numbers. Then, the number in the second line, is the coefficient of w", to express the maximum weight acting by tension upon the diagonal inclining to the right, from the point under that number, and by thrust, upon the diagonal meeting the former at the upper chord.

For instance, the figure 4 in the second line, over d, shows that dl and lf, sustains 4w", the former by tension, and the latter by thrust. This is the weight which must bear at i, in consequence of the weights. at b and d, the only weights that can produce those specific actions upon those members. On the contrary, this action upon dl and f, is only liable to diminution from weight at ƒ, which tends to throw 2w" upon the left abutment through tension of fl and thrust of dl, and consequently diminishes the action upon those members, due to weights at b and d. Therefore, 4w" is the greatest weight sustained by dl and lf, and 2w", the weight sustained by them when the points b, d, and f are loaded, whether the other nodes are loaded or not. There is, however, an alternative in this case, which will be noticed hereafter.

The figure 1 over b, indicates that bn sustains 1w" by tension, and nd the same by thrust, which action is reversed by weights at d and f, which tend to throw 6w" upon these members, namely, 4w", from d, and 2w" from f. Hence, bn is liable to 1w" by tension, and 6w" by thrust, the latter, when d and ƒ are loaded, and b unloaded; and to 5w" (acting by thrust), when all the three points are loaded.

Then, if we form a third series of numbers under the second, by reversing the order of the second, the one series shows the tension, and the other the thrust

to which a member is liable. But as thrust action is not received by any diagonal directly from the weight producing it, but from a tension diagonal meeting it at the upper chord, we do not learn the thrust of a diagonal from the figure over it, at either end, but from the figure over the foot of the diagonal by which the compression is communicated.

Having arranged the diagram as above explained, we form from it a table of greatest weights sustained by the several diagonals, and stresses produced thereby, both of tension and thrust, remembering that tension weights are shown by one series of figures and thrust weights, by the reversed series.

Diagonals. Compression. Tension.

Weights. Stresses. Weights. Stresses.

Under full load.

Weights.

Adding and doubling the several weights, we deduce

the representatives of material.

Under Compression,......

Under Tension,..

The 2 verticals sustain each 12w", giving

The king braces ao and ij, sustain 3w each,

requiring material for the two (62+ 6v)m

=

XLVI. The stress of chords is, as in case of Fig. 12, due to action of obliques, and may be fairly assumed

to be greatest under a full uniform load of the truss. The brace ao has a horizontal thrust = 21w", = tension of ab. The thrust of bn (under the full load), adds

h

5′′ at b, making 26" tension of be. This is in

creased by 9wfor tension of oc, and by 2w" for thrust of cm, making 37w" for tension of cd; and, adding 5w" for tension of dn, and subtracting 2w" for

h

Ο

h

teusion of dl in the opposite direction, we have 40w"! -tension of de.

h

v

We have then, 1 section sustaining 40w" h=40w" h

Making a total stress-208w"-29"acting upon sections of a common length equal to h, and therefore, requiring material represented by 295 M

M.

Upon the upper chord, we have the horizontal action ÿj and fj, producing compression equal to 30w upon jk. Add for horizontal action of ek and kg, 10wh making 40w" stress of kl. Again, add 4w" for horizontal action of lf and ld, and we have 44wh

thrust of lm.

=

ข

Thus, we have for the whole upper chord, 184wh = aggregate stress upon sections of the common length. equal to h. Hence, representative for material =

262M.

Making hv 1, Comp. 45.714м. Ten, 45.714M.

Grand total,

=

.......

91.428M.

We have here a little over 3 per cent less action upon material, than in case of truss Fig. 12, with verticals. The difference is a little less than was shown in my original analysis, that being based on trusses loaded at the upper, and this, at the lower chord; the former giving a trifle more action for the truss with verticals, and a trifle less for the other.

Moreover, the difference was made to show greater still, by assuming that deductions might be made on account of certain diagonals being liable to two kinds of action. For instance, it was supposed that a member formed to sustain a considerable tension stress, might also sustain a small compressive force without additional material (not at the same time, of course), which is undoubtedly the case, on certain occasions; especially in the use of wooden trusses. This would give still greater apparent advantage to truss 13, with regard to economy of material.

XLVII. There is, however, another view as to the action of load upon truss Fig. 13, which may modify the results above shown to a small extent.